According to Woodward, the idea
started with the White House and President Barack Obama.
Republicans voted in majorities for the bill containing
sequestration. Both sides are now blaming one another for letting
it potentially happen.

Woodward detailed a summer of
2011 meeting between Obama and Congressional leaders, Treasury
Secretary Tim
Geithner, and White House national economic
council director Gene Sperling, among others. Woodward paints the
White House as coming up with the idea of a compulsory mechanism
to trigger spending cuts — the sequester — during a summer while
House Speaker John
Boehner was initially "nervous" about it.

Via Woodward:

“A trigger would lock in our commitment,” Sperling explained.
“Even though we disagree on the composition of how to get to the
cuts, it would lock us in. The form of the automatic sequester
would punish both sides. We’d have to September to avert any
sequester”— a legal obligation to make spending cuts.

“Then we could use a medium or big deal to force tax reform,”
Obama said optimistically. “If this is a trigger for tax reform,”
Boehner said, “this could be worth discussing. But as a budget
tool, it’s too complicated. I’m very nervous about this.”

Later in the book, Woodward described Senate Majority Leader
Harry Reid's terrified reaction when White House chief of staff
Jack Lew and director of legislative affairs Rob Nabors told him
about the idea of sequestration:

“We have an idea for the trigger,” Lew said.

“What’s the idea?” Reid asked skeptically. “Sequestration.”

Reid bent down and put his head between his knees, almost as if
he were going to throw up or was having a heart attack. He sat
back up and looked at the ceiling. “A couple of weeks ago,” he
said, “my staff said to me that there is one more possibile
[sic]” enforcement mechanism: sequestration. He said he told
them, “Get the hell out of here. That’s insane. The White House
surely will come up with a plan that will save the day. And you
come to me with sequestration?”

Reid, though, warmed up to the idea when Lew and Nabors told him
that the cuts would be half on defense and half on non-defense
spending. But the thought was that the "super committee" on debt
reduction would be so turned off by the "bomb that no one wanted
to drop" that they would come up with their own spending cuts.
(They did not end up doing that.)

Boehner was skittish, but Republicans eventually agreed to a deal
with the sequester as a trigger for action rather than agree to a
deal with increased revenues.

The deal eventually passed with overwhelming bipartisan
support. 174 of 240 House Republicans supported it,
and Senate Republicans voted for it by a 28-19 margin. The
support from Democrats was lower in the House (a 95-95 split) but
supported it in the Senate 45-6.

In the end, Boehner assured House Republicans that the
sequester would never happen.

“Guys, this would be devastating to Defense,” he said, according
to Woodward. “This would be devastating, from their perspective,
on their domestic priorities. This is never going to happen."